Hat or no hat?

Meredith
I'm pretty militant about putting a hat on my newborn even in the house because we lose a lot of heat from the tops of our heads. However, if I finally get my newborn down to sleep it's a delicate technique trying to put the hat on as she sleeps. Sometimes it wakes her up, sometimes it doesn't. Mind you, our apartment is sometimes drafty for me and I know babies are more sensitive to temp changes. Should I let her be, or should I always put a hat on her with the risk of waking her?